Dot Product and Euclidean Norm

soulrebel934 · **Joined:** Fri, 17 Sep 2010 00:13:04 UTC **Posts:** 17

Supposer that A is a real m x n matrix such that, for any vectors x and y in $\mathbb{R}^n$ , if $x\perp y$ , the $Ax\perp Ay$ . Show that, if x and y are any two vectors in $\mathbb{R}^n$ such that ||x||=||y||, then ||Ax||=||Ay||, where || || denotes the familiar Euclidean norm defined by the dot product. Also show, that if A is not the zero matrix, then $m\ge{n}$ .

Here it was I have so far:

$If{ } \lVert{x}\rVert=\lVert{y}\rVert\implies$
$[((x\cdot{x})^{1/2}= ((y\cdot{y})^{1/2}]^2\implies {\lVert{x}\rVert}^2={\lVert{y}\rVert}^2\implies\lVert{x}\rVert}^2-{\Vert{y}\rVert}^2=0\implies{x\cdot{x}-y\cdot{x}+x\cdot{y}-y\cdot{y}=0\implies(x-y)\cdot(x+y)=0\implies (x-y)\perp(x+y)$

Proving the above statement was part of the hint given in class, but I don't see how that can be used to prove ||Ax||=||Ay||. Also, for the second part I assume m has to be larger because we are in $\mathbb{R}^n$ and multplying a matrix by a vector will result in n-1 dimensions, but is there more to that?

outermeasure · **Posted:** Fri, 22 Oct 2010 09:38:39 UTC

soulrebel934 wrote:

Supposer that A is a real m x n matrix such that, for any vectors x and y in $\mathbb{R}^n$ , if $x\perp y$ , the $Ax\perp Ay$ . Show that, if x and y are any two vectors in $\mathbb{R}^n$ such that ||x||=||y||, then ||Ax||=||Ay||, where || || denotes the familiar Euclidean norm defined by the dot product. Also show, that if A is not the zero matrix, then $m\ge{n}$ .

Here it was I have so far:

$If{ } \lVert{x}\rVert=\lVert{y}\rVert\implies$
$[((x\cdot{x})^{1/2}= ((y\cdot{y})^{1/2}]^2\implies {\lVert{x}\rVert}^2={\lVert{y}\rVert}^2\implies\lVert{x}\rVert}^2-{\Vert{y}\rVert}^2=0\implies{x\cdot{x}-y\cdot{x}+x\cdot{y}-y\cdot{y}=0\implies(x-y)\cdot(x+y)=0\implies (x-y)\perp(x+y)$

Proving the above statement was part of the hint given in class, but I don't see how that can be used to prove ||Ax||=||Ay||. Also, for the second part I assume m has to be larger because we are in $\mathbb{R}^n$ and multplying a matrix by a vector will result in n-1 dimensions, but is there more to that?

(1) Note that all your $\implies$ are actually $\iff$ .

(2) Look at the image of A.

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Dot Product and Euclidean Norm

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